Network Analysis in R
Ted Laderas (laderast@ohsu.edu)
November 5, 2015
Overview
- R Tools for Network Analysis
Network Data Representations useful in R
- graphNEL (node/edge list representation)
- Easy to add/remove nodes and edges
- graphAM (adjacency matrix)
- igraph object (part of the igraph package)
The igraph package
- R, Python and C implementations
- Build and modify networks by nodes and edges
- Extract subGraphs given nodes
- Clique Detection
- Community Detection algorithms (graphlets, random walk, etc)
- Instantiate different background networks
- Erdos-Renyi
- Barabasi-Albert
- Network Metrics
- Centrality (PageRank, Hub)
- Isomorphic Relations (Motif Finding)
What you need to run these slides
I used the following libraries to generate these slides.
require(igraph)
require(knitr)
Loading in Data into igraph
The igraph package has parsers for reading in most of the general file formats for networks. Let’s load in the Karate network from Network Example Data. It’s in GML format, so we’ll need to specify that when we use read_graph().
library(igraph)
karate <- read_graph("data/karate.gml", format="gml")
class(karate)
## [1] "igraph"
Adding Labels and Improving the Layout
That’s not very informative, especially since the original data doesn’t have labels. Let’s add some:
get.vertex.attribute(karate, "id")
## [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
## [24] 24 25 26 27 28 29 30 31 32 33 34
karate <- set.vertex.attribute(karate, "name",
value=c("Chloe", "Emily", "Aaliyah", "Emma",
"Jennifer", "Olivia", "Hannah",
"Jessica", "Sarah", "Lily", "Charlotte",
"Elizabeth", "Abigail", "Rebecca",
"Samantha", "Jacob", "Muhammad", "Shawn",
"Aaron", "Daniel", "Jonah", "Alex",
"Michael", "James", "Ryan", "Jordan",
"Alexander", "Ali", "Tyler", "Kevin",
"Jack", "Ethan", "Luke", "Harry"))
#V() is a way to programmatically access vertices (or nodes) for igraph
V(karate)
## + 34/34 vertices, named:
## [1] Chloe Emily Aaliyah Emma Jennifer Olivia Hannah
## [8] Jessica Sarah Lily Charlotte Elizabeth Abigail Rebecca
## [15] Samantha Jacob Muhammad Shawn Aaron Daniel Jonah
## [22] Alex Michael James Ryan Jordan Alexander Ali
## [29] Tyler Kevin Jack Ethan Luke Harry
#get the ids (as a vector)
V(karate)$name
## [1] "Chloe" "Emily" "Aaliyah" "Emma" "Jennifer"
## [6] "Olivia" "Hannah" "Jessica" "Sarah" "Lily"
## [11] "Charlotte" "Elizabeth" "Abigail" "Rebecca" "Samantha"
## [16] "Jacob" "Muhammad" "Shawn" "Aaron" "Daniel"
## [21] "Jonah" "Alex" "Michael" "James" "Ryan"
## [26] "Jordan" "Alexander" "Ali" "Tyler" "Kevin"
## [31] "Jack" "Ethan" "Luke" "Harry"
## + 78/78 edges (vertex names):
## [1] Chloe --Emily Chloe --Aaliyah Emily --Aaliyah
## [4] Chloe --Emma Emily --Emma Aaliyah --Emma
## [7] Chloe --Jennifer Chloe --Olivia Chloe --Hannah
## [10] Jennifer--Hannah Olivia --Hannah Chloe --Jessica
## [13] Emily --Jessica Aaliyah --Jessica Emma --Jessica
## [16] Chloe --Sarah Aaliyah --Sarah Aaliyah --Lily
## [19] Chloe --Charlotte Jennifer--Charlotte Olivia --Charlotte
## [22] Chloe --Elizabeth Chloe --Abigail Emma --Abigail
## [25] Chloe --Rebecca Emily --Rebecca Aaliyah --Rebecca
## [28] Emma --Rebecca Olivia --Muhammad Hannah --Muhammad
## + ... omitted several edges
Graph Attributes
Each element of a graph (vertices and edges) can be customized by passing in the appropriate attribute. For more info, check out http://www.inside-r.org/packages/cran/igraph/docs/attributes.
Some properties are automatically mapped, such as V(graph)$color or E(graph)$color. If you add a new attribute, you’ll need to map this to a plotting property in plot.igraph()
Remember, that after changing graph properties, such as node size, you need to run your layout_ function again.
#setting node properties
#can also use set.vertex.attributes() here
newKarate <- karate
faction <- c(1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2,
1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2)
V(newKarate)$color <- faction
#Set edge attributes using E(accessor)
#can pass a named list for each node as well
#You can also use set.edge.attributes() here
E(newKarate)$color <- "red"
#Here we instantiate a weight vector using sample
weightvec <- sample(c(1,2,3,4), length(E(newKarate)), replace = TRUE)
#Name each edge
names(weightvec) <- E(newKarate)$name
weightvec
## [1] 4 2 4 1 4 3 3 3 2 4 3 1 1 2 3 3 3 3 2 2 3 3 2 4 1 4 4 1 4 2 4 4 3 1 4
## [36] 4 2 4 4 4 2 2 1 2 2 2 4 3 2 4 2 4 4 2 2 2 1 1 2 2 2 1 3 2 1 2 2 4 1 1
## [71] 2 4 2 4 4 2 3 2
E(newKarate)$weight <- weightvec
#note we need to map an attribute to a property of the graph (igraph is dumb)
plot(newKarate, edge.width=E(karate)$weight)
Simple Measures
Degree Distribution and Average Path Length
## Chloe Emily Aaliyah Emma Jennifer Olivia Hannah
## 16 9 10 6 3 4 4
## Jessica Sarah Lily Charlotte Elizabeth Abigail Rebecca
## 4 5 2 3 1 2 5
## Samantha Jacob Muhammad Shawn Aaron Daniel Jonah
## 2 2 2 2 2 3 2
## Alex Michael James Ryan Jordan Alexander Ali
## 2 2 5 3 3 2 4
## Tyler Kevin Jack Ethan Luke Harry
## 3 4 4 6 12 17
sort(degree(karate), decreasing = TRUE)
## Harry Chloe Luke Aaliyah Emily Emma Ethan
## 17 16 12 10 9 6 6
## Sarah Rebecca James Olivia Hannah Jessica Ali
## 5 5 5 4 4 4 4
## Kevin Jack Jennifer Charlotte Daniel Ryan Jordan
## 4 4 3 3 3 3 3
## Tyler Lily Abigail Samantha Jacob Muhammad Shawn
## 3 2 2 2 2 2 2
## Aaron Jonah Alex Michael Alexander Elizabeth
## 2 2 2 2 2 1
average.path.length(karate)
## [1] 2.4082
shortPaths <- get.shortest.paths(karate, from="Muhammad")
plot(karate, mark.groups=shortPaths$vpath[c(10)])
Drunken Stumbling
Let’s generate a random walk from Jennifer and see where we get.
w <- random_walk(karate, start = "Jennifer", steps =1000)
#who is the most visited?
sort(table(w$name), decreasing = TRUE)
##
## Chloe Harry Aaliyah Emily Luke Emma Ethan
## 120 106 64 61 56 43 42
## Sarah Olivia Jack Daniel Rebecca Hannah James
## 37 34 30 29 28 27 26
## Jessica Jordan Charlotte Ryan Abigail Jennifer Kevin
## 23 20 19 19 18 18 18
## Muhammad Tyler Alex Ali Aaron Samantha Alexander
## 17 16 15 15 14 14 13
## Shawn Elizabeth Lily Jacob Michael Jonah
## 12 11 11 9 8 7
#normalize table by number of visits
probKarate <- table(w$name)/1000
Influencers
Who are the influencers (or hubs) in this social network? Let’s use some network centrality measures to see. Are there any differences?
Who gives the highest correlation with our drunken walk probabilities?
pr <- page.rank(karate)
sort(pr$vector, decreasing = TRUE)
## Harry Chloe Luke Aaliyah Emily Ethan
## 0.100919182 0.096997285 0.071693226 0.057078509 0.052876924 0.037158087
## Emma James Sarah Rebecca Olivia Hannah
## 0.035859858 0.031522515 0.029766056 0.029536456 0.029111155 0.029111155
## Kevin Ali Jack Jessica Jennifer Charlotte
## 0.026288538 0.025639767 0.024590155 0.024490497 0.021977952 0.021977952
## Ryan Jordan Daniel Tyler Muhammad Alexander
## 0.021076034 0.021006197 0.019604636 0.019573459 0.016784005 0.015044038
## Abigail Shawn Alex Samantha Jacob Aaron
## 0.014644892 0.014558677 0.014558677 0.014535994 0.014535994 0.014535994
## Jonah Michael Lily Elizabeth
## 0.014535994 0.014535994 0.014309397 0.009564745
nk <- V(karate)$name
cor(pr$vector[nk], probKarate[nk])
## [1] 0.9691531
ec <- eigen_centrality(karate)
sort(ec$vector, decreasing = TRUE)
## Harry Chloe Aaliyah Luke Emily Sarah
## 1.00000000 0.95213237 0.84955420 0.82665886 0.71233514 0.60906844
## Rebecca Emma Ethan Jack Jessica James
## 0.60657439 0.56561431 0.51165649 0.46806481 0.45789093 0.40207086
## Daniel Kevin Ali Tyler Lily Jonah
## 0.39616224 0.36147301 0.35749923 0.35107297 0.27499812 0.27159396
## Michael Jacob Aaron Samantha Shawn Alex
## 0.27159396 0.27159396 0.27159396 0.27159396 0.24747879 0.24747879
## Abigail Olivia Hannah Jennifer Charlotte Alexander
## 0.22566382 0.21288383 0.21288383 0.20347148 0.20347148 0.20242852
## Jordan Ryan Elizabeth Muhammad
## 0.15857597 0.15280670 0.14156633 0.06330461
cor(ec$vector[nk], probKarate[nk])
## [1] 0.8763587
hs <- hub_score(karate)
sort(hs$vector, decreasing = TRUE)
## Harry Chloe Aaliyah Luke Emily Sarah
## 1.00000000 0.95213237 0.84955420 0.82665886 0.71233514 0.60906844
## Rebecca Emma Ethan Jack Jessica James
## 0.60657439 0.56561431 0.51165649 0.46806481 0.45789093 0.40207086
## Daniel Kevin Ali Tyler Lily Samantha
## 0.39616224 0.36147301 0.35749923 0.35107297 0.27499812 0.27159396
## Jacob Aaron Jonah Michael Shawn Alex
## 0.27159396 0.27159396 0.27159396 0.27159396 0.24747879 0.24747879
## Abigail Olivia Hannah Jennifer Charlotte Alexander
## 0.22566382 0.21288383 0.21288383 0.20347148 0.20347148 0.20242852
## Jordan Ryan Elizabeth Muhammad
## 0.15857597 0.15280670 0.14156633 0.06330461
cor(hs$vector[nk], probKarate[nk])
## [1] 0.8763587
Cliques?
Who’s the most connected groups to each other? Enquiring minds want to know…
##get all cliques in the data
karateCliques <- cliques(karate)
length(karateCliques)
## [1] 170
#let's try finding the largest cliques
largeKarateCliques <- largest_cliques(karate)
largeKarateCliques
## [[1]]
## + 5/34 vertices, named:
## [1] Jessica Chloe Emily Aaliyah Emma
##
## [[2]]
## + 5/34 vertices, named:
## [1] Emma Chloe Emily Aaliyah Rebecca
What are the Karate communities?
Can we split the network up into smaller communities? There are number of community detection algorithms we can use to see if we can cleanly separate our network into smaller modules, or communities.
##Segment the karate network using the fast-greedy algorithm
fgComm <- fastgreedy.community(karate)
plot(fgComm, karate)
##Segment the karate network using the walktrap algorithm
wkComm <- walktrap.community(karate)
plot(wkComm, karate)
##how close are the algorithms?
compare(membership(fgComm), membership(wkComm))
## [1] 1.14204
Other searches
How do we find all of Emma’s friends?
emmasNeighbors <- neighbors(karate, v="Emma", mode="all")
emmaSubnet <- c("Emma",emmasNeighbors$name)
emmaSubnet
## [1] "Emma" "Chloe" "Emily" "Aaliyah" "Jessica" "Abigail" "Rebecca"
emmaSubgraph <- induced.subgraph(karate, emmaSubnet)
plot(emmaSubgraph)
Other Plot Layouts
igraph’s built in plotting is not super customizable, but you can change layouts. All of the different igraph layouts begin with “layout_”.
#circle plot
plot(karate, layout=layout_in_circle)
#layout plot on grid
plot(karate, layout=layout_on_grid)
#force-directed layout for large graphs
plot(karate, layout=layout_with_lgl)
#layout plot with center (star layout)
plot(karate, layout=layout_as_star)
#layout plot on grid
plot(karate, layout=layout_with_dh)
Generating graphs from scratch
Oftentimes, you will need to generate a bunch of graphs from scratch, for a background distribution or something. There are some built in functions to do that.
#Random Graphs
rgGraph <- random.graph.game(34, p.or.m = 0.2)
plot(rgGraph)
#Erdos Renyi Graphs
erGraph <- erdos.renyi.game(34, 0.1)
plot(erGraph)
#Barabasi graphs - power law distributed
bgGraph <- barabasi.game(34, 1)
plot(bgGraph)
#Confirm edge distribution is correct
hist(degree(bgGraph))
Converting igraph Objects to Other Representations
The graph package (not the igraph package!) contains the graphNEL and graphAM representations. I don’t really go into using them here, but just know that you can convert back and forth from these different representations using different commands.
##
## Attaching package: 'graph'
##
## The following objects are masked from 'package:igraph':
##
## degree, edges, intersection
#graphNEL objects
karateGN <- igraph.to.graphNEL(karate)
karateGN
## A graphNEL graph with undirected edges
## Number of Nodes = 34
## Number of Edges = 78
#Adjacency matrix
karateAM <- as(karateGN, "graphAM")
karateAM
## A graphAM graph with undirected edges
## Number of Nodes = 34
## Number of Edges = 78
#look at the actual adjacency matrix
karateAdjMat <- karateAM@adjMat
rownames(karateAdjMat) <- colnames(karateAdjMat)
rowSums(karateAdjMat)
## Chloe Emily Aaliyah Emma Jennifer Olivia Hannah
## 16 9 10 6 3 4 4
## Jessica Sarah Lily Charlotte Elizabeth Abigail Rebecca
## 4 5 2 3 1 2 5
## Samantha Jacob Muhammad Shawn Aaron Daniel Jonah
## 2 2 2 2 2 3 2
## Alex Michael James Ryan Jordan Alexander Ali
## 2 2 5 3 3 2 4
## Tyler Kevin Jack Ethan Luke Harry
## 3 4 4 6 12 17
## Chloe Emily Aaliyah Emma Jennifer Olivia Hannah
## 16 9 10 6 3 4 4
## Jessica Sarah Lily Charlotte Elizabeth Abigail Rebecca
## 4 5 2 3 1 2 5
## Samantha Jacob Muhammad Shawn Aaron Daniel Jonah
## 2 2 2 2 2 3 2
## Alex Michael James Ryan Jordan Alexander Ali
## 2 2 5 3 3 2 4
## Tyler Kevin Jack Ethan Luke Harry
## 3 4 4 6 12 17
## Chloe Emily Aaliyah Emma Jennifer Olivia Hannah Jessica Sarah
## Chloe 0 1 1 1 1 1 1 1 1
## Emily 1 0 1 1 0 0 0 1 0
## Aaliyah 1 1 0 1 0 0 0 1 1
## Emma 1 1 1 0 0 0 0 1 0
## Jennifer 1 0 0 0 0 0 1 0 0
## Olivia 1 0 0 0 0 0 1 0 0
## Hannah 1 0 0 0 1 1 0 0 0
## Jessica 1 1 1 1 0 0 0 0 0
## Sarah 1 0 1 0 0 0 0 0 0
## Lily 0 0 1 0 0 0 0 0 0
## Charlotte 1 0 0 0 1 1 0 0 0
## Elizabeth 1 0 0 0 0 0 0 0 0
## Abigail 1 0 0 1 0 0 0 0 0
## Rebecca 1 1 1 1 0 0 0 0 0
## Samantha 0 0 0 0 0 0 0 0 0
## Jacob 0 0 0 0 0 0 0 0 0
## Muhammad 0 0 0 0 0 1 1 0 0
## Shawn 1 1 0 0 0 0 0 0 0
## Aaron 0 0 0 0 0 0 0 0 0
## Daniel 1 1 0 0 0 0 0 0 0
## Jonah 0 0 0 0 0 0 0 0 0
## Alex 1 1 0 0 0 0 0 0 0
## Michael 0 0 0 0 0 0 0 0 0
## James 0 0 0 0 0 0 0 0 0
## Ryan 0 0 0 0 0 0 0 0 0
## Jordan 0 0 0 0 0 0 0 0 0
## Alexander 0 0 0 0 0 0 0 0 0
## Ali 0 0 1 0 0 0 0 0 0
## Tyler 0 0 1 0 0 0 0 0 0
## Kevin 0 0 0 0 0 0 0 0 0
## Jack 0 1 0 0 0 0 0 0 1
## Ethan 1 0 0 0 0 0 0 0 0
## Luke 0 0 1 0 0 0 0 0 1
## Harry 0 0 0 0 0 0 0 0 1
## Lily Charlotte Elizabeth Abigail Rebecca Samantha Jacob Muhammad
## Chloe 0 1 1 1 1 0 0 0
## Emily 0 0 0 0 1 0 0 0
## Aaliyah 1 0 0 0 1 0 0 0
## Emma 0 0 0 1 1 0 0 0
## Jennifer 0 1 0 0 0 0 0 0
## Olivia 0 1 0 0 0 0 0 1
## Hannah 0 0 0 0 0 0 0 1
## Jessica 0 0 0 0 0 0 0 0
## Sarah 0 0 0 0 0 0 0 0
## Lily 0 0 0 0 0 0 0 0
## Charlotte 0 0 0 0 0 0 0 0
## Elizabeth 0 0 0 0 0 0 0 0
## Abigail 0 0 0 0 0 0 0 0
## Rebecca 0 0 0 0 0 0 0 0
## Samantha 0 0 0 0 0 0 0 0
## Jacob 0 0 0 0 0 0 0 0
## Muhammad 0 0 0 0 0 0 0 0
## Shawn 0 0 0 0 0 0 0 0
## Aaron 0 0 0 0 0 0 0 0
## Daniel 0 0 0 0 0 0 0 0
## Jonah 0 0 0 0 0 0 0 0
## Alex 0 0 0 0 0 0 0 0
## Michael 0 0 0 0 0 0 0 0
## James 0 0 0 0 0 0 0 0
## Ryan 0 0 0 0 0 0 0 0
## Jordan 0 0 0 0 0 0 0 0
## Alexander 0 0 0 0 0 0 0 0
## Ali 0 0 0 0 0 0 0 0
## Tyler 0 0 0 0 0 0 0 0
## Kevin 0 0 0 0 0 0 0 0
## Jack 0 0 0 0 0 0 0 0
## Ethan 0 0 0 0 0 0 0 0
## Luke 0 0 0 0 0 1 1 0
## Harry 1 0 0 0 1 1 1 0
## Shawn Aaron Daniel Jonah Alex Michael James Ryan Jordan
## Chloe 1 0 1 0 1 0 0 0 0
## Emily 1 0 1 0 1 0 0 0 0
## Aaliyah 0 0 0 0 0 0 0 0 0
## Emma 0 0 0 0 0 0 0 0 0
## Jennifer 0 0 0 0 0 0 0 0 0
## Olivia 0 0 0 0 0 0 0 0 0
## Hannah 0 0 0 0 0 0 0 0 0
## Jessica 0 0 0 0 0 0 0 0 0
## Sarah 0 0 0 0 0 0 0 0 0
## Lily 0 0 0 0 0 0 0 0 0
## Charlotte 0 0 0 0 0 0 0 0 0
## Elizabeth 0 0 0 0 0 0 0 0 0
## Abigail 0 0 0 0 0 0 0 0 0
## Rebecca 0 0 0 0 0 0 0 0 0
## Samantha 0 0 0 0 0 0 0 0 0
## Jacob 0 0 0 0 0 0 0 0 0
## Muhammad 0 0 0 0 0 0 0 0 0
## Shawn 0 0 0 0 0 0 0 0 0
## Aaron 0 0 0 0 0 0 0 0 0
## Daniel 0 0 0 0 0 0 0 0 0
## Jonah 0 0 0 0 0 0 0 0 0
## Alex 0 0 0 0 0 0 0 0 0
## Michael 0 0 0 0 0 0 0 0 0
## James 0 0 0 0 0 0 0 0 1
## Ryan 0 0 0 0 0 0 0 0 1
## Jordan 0 0 0 0 0 0 1 1 0
## Alexander 0 0 0 0 0 0 0 0 0
## Ali 0 0 0 0 0 0 1 1 0
## Tyler 0 0 0 0 0 0 0 0 0
## Kevin 0 0 0 0 0 0 1 0 0
## Jack 0 0 0 0 0 0 0 0 0
## Ethan 0 0 0 0 0 0 0 1 1
## Luke 0 1 0 1 0 1 1 0 0
## Harry 0 1 1 1 0 1 1 0 0
## Alexander Ali Tyler Kevin Jack Ethan Luke Harry
## Chloe 0 0 0 0 0 1 0 0
## Emily 0 0 0 0 1 0 0 0
## Aaliyah 0 1 1 0 0 0 1 0
## Emma 0 0 0 0 0 0 0 0
## Jennifer 0 0 0 0 0 0 0 0
## Olivia 0 0 0 0 0 0 0 0
## Hannah 0 0 0 0 0 0 0 0
## Jessica 0 0 0 0 0 0 0 0
## Sarah 0 0 0 0 1 0 1 1
## Lily 0 0 0 0 0 0 0 1
## Charlotte 0 0 0 0 0 0 0 0
## Elizabeth 0 0 0 0 0 0 0 0
## Abigail 0 0 0 0 0 0 0 0
## Rebecca 0 0 0 0 0 0 0 1
## Samantha 0 0 0 0 0 0 1 1
## Jacob 0 0 0 0 0 0 1 1
## Muhammad 0 0 0 0 0 0 0 0
## Shawn 0 0 0 0 0 0 0 0
## Aaron 0 0 0 0 0 0 1 1
## Daniel 0 0 0 0 0 0 0 1
## Jonah 0 0 0 0 0 0 1 1
## Alex 0 0 0 0 0 0 0 0
## Michael 0 0 0 0 0 0 1 1
## James 0 1 0 1 0 0 1 1
## Ryan 0 1 0 0 0 1 0 0
## Jordan 0 0 0 0 0 1 0 0
## Alexander 0 0 0 1 0 0 0 1
## Ali 0 0 0 0 0 0 0 1
## Tyler 0 0 0 0 0 1 0 1
## Kevin 1 0 0 0 0 0 1 1
## Jack 0 0 0 0 0 0 1 1
## Ethan 0 0 1 0 0 0 1 1
## Luke 0 0 0 1 1 1 0 1
## Harry 1 1 1 1 1 1 1 0
Exporting out graphs
Sometimes you want to take your subgraphs and use a program like Cytoscape to make the visualization prettier.
The GML format will preserve most of your information.
cyREST API
The cyREST API allows you to programatically transfer network data into Cytoscape. It’s a bit quirky to set up and use, but once you set it up, you can transfer all network attributes (edge, node, and graph) that you’ve assigned to your GraphNEL objects directly into Cytoscape.
For more info, check this link out: https://github.com/idekerlab/cy-rest-R
Rgraphviz
- Graphviz: mature package for network and graph visualization (AT&T)
- Can define custom node graphics using viewports
- Multiple Layouts
- dot (hierarchical)
- neato (force-directed, spring-model)
- twopi (radial plots)
- circo (circular graphs)
- Clusters (subGraphs) for more control over the layout
Custom Node Graphics + KEGGgraph + cluster layout in Rgraphviz
I used a combination of ggplot2 and Rgraphviz to plot time-series data on a pathway defined by KEGG. To get this to work, I had to define some clusters in the graph layout.
Drawbacks to Rgraphviz
Graphviz is not guaranteed to give you the best layout - it can take a lot of customization and specification of clusters. You may want to take your networks into a tool such as Cytoscape that will let you customize your graph much more.
SessionInfo
## R version 3.1.3 (2015-03-09)
## Platform: x86_64-apple-darwin13.4.0 (64-bit)
## Running under: OS X 10.10.3 (Yosemite)
##
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] graph_1.44.1 knitr_1.10.5 igraph_1.0.1
##
## loaded via a namespace (and not attached):
## [1] BiocGenerics_0.12.1 digest_0.6.8 evaluate_0.7
## [4] formatR_1.2 htmltools_0.2.6 magrittr_1.5
## [7] parallel_3.1.3 rmarkdown_0.7 stats4_3.1.3
## [10] stringi_0.5-5 stringr_1.0.0 tools_3.1.3
## [13] yaml_2.1.13
